The objective is to give the apparently relevant background to the long-standing conjecture that there are no projective planes of order \(n\) unless \(n\) is a prime power.

The material still to come is:
(1) Pappus implies Desargues (this post);
(2) Wedderburn’s theorem
(3) Desargues implies the plane is over a division ring
(4) Pappus implies the plane is over a field
(5) Stinson’s proof that there are no orthogonal Latin squares of order 6
(6) the construction of non-Desarguesian planes
(7) Lee’s proof that there are two orthogonal Latin squares for any order \(n\gt 6\)

Here it is important that the point \(i\) does not lie on any of the lines apart from those listed, except possibly \(L_i\). Again if the line \(L_i\) is removed, then it can be recovered from the other 9.

So suppose that \(L_0\) is removed from the Desargues list. That leaves us with:

Take: point 10 to be the intersection of \(L_3,L_6\); line \(L_{10}\) to be the line through points 3,6; point 11 to be the intersection of \(L_7,L_{10}\); line \(L_{11}\) to be the line through points 0,10; point 12 to be the intersection of \(L_2,L_{11}\); and point 13 to be the intersection of \(L_1,L_{11}\). That takes us to: